PHYSICAL REVIEW LETTERS
PRL 101, 023601 (2008)
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Piecewise Adiabatic Population Transfer in a Molecule via a Wave Packet
Evgeny A. Shapiro,1 Avi Pe’er,3 Jun Ye,3 and Moshe Shapiro1,2,4
1
Department of Chemistry, The University of British Columbia, Vancouver, V6T 1Z2, Canada
2
Department of Physics, The University of British Columbia, Vancouver, V6T 1Z2, Canada
3
JILA, National Institute of Standards and Technology and University of Colorado, Boulder, Colorado 80309-0440, USA
4
Department of Chemical Physics, The Weizmann Institute, Rehovot, 76100, Israel
(Received 26 October 2007; published 7 July 2008)
We propose a class of schemes for robust population transfer between quantum states that utilize trains
of coherent pulses, thus forming a generalized adiabatic passage via a wave packet. We study piecewise
stimulated Raman adiabatic passage with pulse-to-pulse amplitude variation, and piecewise chirped
Raman passage with pulse-to-pulse phase variation, implemented with an optical frequency comb. In
the context of production of ultracold ground-state molecules, we show that with almost no knowledge of
the excited potential, robust high-efficiency transfer is possible.
DOI: 10.1103/PhysRevLett.101.023601
PACS numbers: 42.50.Hz, 33.80.b, 82.50.Nd, 82.53.Kp
With the advent of technologies that deal with quantum
properties of matter, there is a need for methods of controlling quantum systems that are, on one hand, robust and
conceptually simple, and on the other, flexible and applicable far beyond few-level arrangements. To address this
need, our goal in this work is to utilize the intuition and
control recipes of simple schemes for robust transfer of
quantum population based on adiabatic passage (AP), and
apply them to many-level quantum systems with complex
dynamics. The key components of the method are coherent
accumulation of amplitudes by a train of phase coherent
optical pulses, and interference of quantum pathways in
multistate population transfer.
We test our ideas, simulating dynamics in KRb molecules, which is of special interest because of the prospect
for creating deeply bound ultracold polar molecules starting from loosely bound Feshbach states [1,2]. As translationally and vibrationally cold polar molecules are a
prerequisite for many applications [3], their robust highyield creation is a forefront goal.
Coherent accumulation has been studied in the context
of perturbative control [4] and later on in precision spectroscopy and high resolution quantum control using frequency combs [5]. Its use for complete population transfers
was studied so far twice. One was the analysis within the 3
level model of piecewise adiabatic passage (PAP) [6]; the
other for population transfer through a wave packet with a
coherent train of pump-dump pulses [7]. Here, we unite the
two as special cases within one general framework, where
the combination of adiabatic transfer concepts with a coherent train excitation (optical frequency comb) preserves
the robustness of adiabatic transfer, but is applicable to
composite quantum states.
The concept of piecewise adiabaticity is elucidated with
two main examples. In piecewise stimulated Raman adiabatic passage (STIRAP), robust transfer is achieved
through a slow variation of the intensity envelope of the
driving pulse trains. In the other, which we term ‘‘piece0031-9007=08=101(2)=023601(4)
wise chirped Raman passage’’ (CRP), robustness is obtained by slow variation of the excitation phase. We
demonstrate how both examples can be extended to the
case of the intermediate state being a wave packet, i.e., a
multiplicity of intermediate states. In the spectral domain,
the transfer is seen as a constructive interference of several
AP pathways [8,9]. Last, we discuss the case of transfer
without detailed knowledge of the intermediate dynamics.
We show that by scanning the comb parameters of a train of
unshaped pulse pairs (repetition rate and intrapair time
separation) it is possible to achieve high-efficiency transfer. The intermediate dynamics can then be studied using a
two-dimensional mapping of the transfer efficiency as a
function of the comb parameters.
A traditional AP transfers population from an initial to a
target state by dressing the quantum system in slowly
changing fields. If one of the time-dependent field-dressed
eigenstates coincides with the initial state at the beginning
of the process, and with the target at the end [9–12], a
complete and robust transfer is obtained as long as the
dressing fields change adiabatically. Accordingly, the prescription of PAP [6] can be summarized as follows: For
reference, consider any traditional AP, driven by slowly
varying fields. Then, break the reference AP into a set of
time intervals n , so short that only a small fraction of
population is transferred between the eigenstates during
each interval. Within each interval, the driving fields can be
replaced by any other field amplitude shape, as long as the
integral action of each of the fields over n remains the
same. For example, the smooth reference field can be
replaced by a train of mutually coherent pulses. Last,
vary the interpulse ‘‘silent’’ time of the system’s free
evolution. Controlling the silent, free evolution durations
and the phase of each of the driving fields allows control
over the arising Ramsey-type interference picture.
Reference [6] introduced piecewise versions of STIRAP
[11,12], which transfers population in a three-level system
from state j1i into state j3i via the intermediate j2i. In the
023601-1
© 2008 The American Physical Society
2
0
P;D n P;D n n0 =2
(1)
as shown in Fig. 1(b). Pulse trains with such a piecewise
chirp demonstrated piecewise adiabatic following in a twostate system [13]. Here we observe that the piecewise CRP
in a three-level system is robust with respect to the field
strengths, the number of pulse pairs, the values and signs of
P and D , and to additional delays between the pump and
dump pulse trains.
In both foregoing examples, the details of the single
pulse dynamics are not important, only the values of the
basis state amplitudes after each n matter. Based on this
observation, below we extend the conceptual scheme of
PAP, generalizing the eigenstate j2i to a multitude of states,
forming a wave packet. This wave packet undergoes complex dynamics between the pairs of pump and dump kicks,
but if the pulse repetition time coincides with a revival, the
wave packet (almost) returns to its original state by the
time the next pair of kicks arrives. The kicks, in turn,
become rather complex operations implemented via an
interplay of the shaped femtosecond pulses with the intermediate wave packet dynamics. We expect that, as long as
the action of the kicks on the basis states j1i, j2i, and j3i
mimics that of the dressing fields in the reference AP
coarse-grained over n , the enforced population transfers
in both schemes is insensitive to the parameters of the
Field strength (10 6 V/cm)
Phase
10
(a)
Field strength (10 5 V/cm)
(b)
5
2
0.6
0.8
1
1.2
Time (ps)
1.4
1.6
0
6
0
0
Time (ps)
driving fields. Further, high fidelity of the revivals turns
out to be unnecessary, as we demonstrate later with unshaped pulses.
Below we study this general idea simulating creation of
deeply bound X1 KRb molecules from a a3 state.
The molecular potentials used in the simulations are shown
schematically in Fig. 2. Eigenenergies and transition
strengths are calculated using the algorithm FDEXTR [14]
with the data input from Refs. [15,16]. The excited electronic state is modeled as the LS-coupled A1 and b3 potentials. Although the model is too simple to precisely
describe molecular dynamics on submicrosecond time
scales, it retains the main complexity of the problem due
to the coupled singlet-triplet evolution. We choose j1i to be
the vibrational state v 5 (E 157 cm1 ) of the a3 potential (Fig. 2). To trace the fidelity of the method, we
also consider the amplitudes of vibrational states around
the input state (v 0; 1; . . . ; 12 of the a3 potential).
The wave packet j2i is composed of up to 20 vibrational
states of the LS-coupled A1 b3 potentials with the
energies E 11 000–11 400 cm1 . State j3i is v 22
(E 2490 cm1 ) of the ground X1 potential; the
neighboring X1 states v 16; . . . ; 28 are included in
the simulations as well.
The system, initially in state j1i, is driven by a series of
mutually coherent pump and dump femtosecond pulses. A
single pump pulse [110 fs FWHM in intensity,
sin2 t shaped, as shown in Fig. 3(a)], excites an A1 b3 wave packet, which oscillates on the coupled A1 b3 potential surfaces with the singlet and triplet vibrational periods of the order of 800 –900 fs. After several
vibrations the dynamics become irregular. However, at
certain times, which correspond to two-surface revivals,
the wave packet rephases with a fidelity reaching 0:9. In
our simulations, high-fidelity revivals were observed,
roughly, every 80 ps; the particular revival time is sensitive
to the spectroscopic data used as input. Following its
excitation, the wave packet can be dumped into the single
vibrational level j3i. The shape of the femtosecond dump
pulse is found by requiring that the wave packet before
dumping overlaps well with the wave packet that would
have been excited by the time-reversed dump, in a manner
dynamics
Potential energy
key example, the field was given as a train of femtosecond
‘‘pump’’ and ‘‘dump’’ pulse pairs, resonantly coupling the
eigenstates j1i and j2i (pump), and the eigenstates j2i and
j3i (dump). The pulses—‘‘kicks’’—did not overlap in
time and the pulse train envelopes were slowly varied to
achieve PAP, as shown in Fig. 1(a).
Another conventional AP in a three-level system is CRP
[11]. The pump and dump pulses share a similar smooth
temporal intensity profile; however, both pulses are frequency chirped, so that their phases are P;D t !P;D t !
t t0 2 =2. Like STIRAP, CRP transfers population
P;D
from state j1i into j3i in a robust manner; unlike STIRAP,
CRP creates transient population in state j2i. For the piecewise version, all pulses are exactly on resonance. The
effect of chirp is mimicked by varying the carrier phase
of each pulse 0
P;D n P;D n; t !P;D t quadratically
with the pulse number n:
4
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PHYSICAL REVIEW LETTERS
PRL 101, 023601 (2008)
dump:
shaped
laser
pulses
X1 Σ +
FIG. 1 (color online). Sequences of femtosecond pulses in
piecewise STIRAP (a) and piecewise CRP (b) between states
j1i 3S, j2i 4P, and j3i 5S of Na [19]. Dashed blue: pump
field envelope; solid red: dump field envelope; gray: the carrier
phase in piecewise CRP with P D 0:2.
|2
>
b3Π
A1Σ
pump:
laser pulses
+
a3Σ
|1
>
|3
>
Internuclear distance
FIG. 2 (color online). KRb potentials, basis states, and PAP
arrangements discussed in the text.
023601-2
PHYSICAL REVIEW LETTERS
PRL 101, 023601 (2008)
Field strength
Field strength
1
(b) 5
(a)
0
0
-5
-1
0
-3
0
-3
3
Time (ps)
3
Time (ps)
FIG. 3 (color online). Time-dependent field of the pump (a),
and dump (b) pulses. The field in each pulse is normalized
relative to the maximum of the pump field.
similar to the prescription of Ref. [7]. The thus-found
dump field is shown in Fig. 3(b).
The simulation results for both piecewise STIRAP and
piecewise CRP are presented in Fig. 4. For STIRAP, shown
in Figs. 4(a) and 4(b), the field consists of 200 pairs of
pump and dump pulses, all shaped as in Fig. 3. The train
envelopes vary linearly on the pulse number [Fig. 4(a)].
The interpair separation is 1310.59 ps, which is close to a
revival time of the wave packet j2i. The carrier frequencies
are chosen to match the Raman condition !Raman !P !D . The carrier optical phase is kept constant throughout
each pulse train by choosing the carrier frequency to
coincide with a comb tooth [6]:
!P;D 2NP;D frep f0P;D ;
(2)
where frep is the common repetition rate and f0P;D is the
carrier-envelope offset frequency in the pump (dump)
train. The process is interferometrically sensitive to the
interpulse delay (frep ) and requires phase locking of the
Field scaling (105 V/cm)
5
Phase
(a)
0
Field scaling (106 V/cm)
4
(c)
0
0
Populations
1
a3Σ+
2
Populations
(b)
(d)
a3Σ+
1
0
0
0.02 A1Σ−b3Π
A1Σ−b3Π
0
0.1
0
0.7 X1Σ+
X1Σ+
0.5
0
Compressed time
Compressed time
0
FIG. 4 (color online). Dynamics of the STIRAP-like (a),(b)
and CRP-like (c),(d) PAP in KRb. (a),(c) Envelopes of the pump
(dashed blue) and dump (solid red) pulses, and the carrier phase
in piecewise CRP [gray in (c)]. Each point shows the factor by
which the field shown in Fig. 3 is scaled. The carrier phase 0
in the piecewise CRP, same for the pump and dump, is shown by
the gray staircase in (c). (b),(d) Population dynamics of all the
levels in the system. Time is given at the ‘‘compressed’’ set of
points: output is shown only for the times when the field is on.
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pump and dump combs; however, phase stabilization of
frequency combs is now a standard technique [5].
If the spontaneous decay of the A1 b3 states is
neglected, the calculation predicts transferring 92% of
population from j1i into j3i. Introducing a 15-ns A1 b3 decay time reduces the transfer efficiency to 75%.
The maximum combined transient population of all the
A1 b3 levels reaches 0:04. The transfer efficiency
is found to be rather insensitive to the field strength profiles
in the pulse trains, and varies within about 15% if the
number of pulses is varied between 35 and 300.
Figures 4(c) and 4(d) show an example of CRP-like PAP.
The field consists of 20 pairs of pump and dump pulses.
The pump and dump pulse train envelopes are Gaussian
[Fig. 4(c)]. The interpair separation is 1309.93 ps. The
central carrier phase in each field evolves with the pulse
number according to Eq. (1), with P D 0:09. The
calculation with decaying A1 b3 levels predicts
transferring 60% of the initial population into the target
state; neglecting the decay raises this number to 65%.
Approximately 2% of the population ends up distributed
among eigenstates other than j3i of the target X1 manifold; 22% of the initial population is transferred to neighboring states of j1i in the a3 manifold at the end of the
process. Unlike in the simple 3-level case, CRP-like transfer in the molecule is found to be significantly more
sensitive than the STIRAP-like transfer to the number of
pulses and the intensity profiles of the pump and dump
pulse trains. We note that the required train envelope
modulations (either in amplitude or phase) can be readily
generated experimentally by present modulators for standard 1–10 ns repetition-period pulse sources.
The dependence of the transfer efficiency on T in both
STIRAP-like and CRP-like PAP is irregular due to the
complex dynamics of the A1 b3 wave packet.
Viewed from the spectral perspective, each of the A1 b3 eigenstates can serve as an intermediate level for a
separate AP pathway. Reminiscent of the conventional
coherent control via a wave packet [9,17], the AP pathways
interfere to provide the population transfer into the target
state. Thus the frequency comb source drives a coherently
controlled AP process [5,8] via the set of intermediate
levels. Matching the repetition time and carrier phase to
the wave packet revival time and phase ensures that all the
possible AP pathways participate; spectral shaping of the
pump and dump pulses sets constructive interference
among the pathways.
Tailoring the optimal pulses requires detailed knowledge
of the excited state structure, which is not always available.
But what happens if the pulses are not shaped at all?
Figure 5(a) investigates the total efficiency of the
STIRAP-like transfer under both pump and dump trains
consisting of 50 sin2 t-shaped pulses, with respect to the
interpair delay T 1=frep and intrapair delay T between the pump and dump pulses. Raman resonance is
maintained throughout the frep scan by varying the
023601-3
PHYSICAL REVIEW LETTERS
PRL 101, 023601 (2008)
1.5
(a)
Scale
δT (ps)
1
-2
1309
(c)
P( )
(b)
8
0
-2
δT (ps)
1.5
0
ω (cm -1)
Spectrum
1
0
1311
∆T (ps)
90
FIG. 5 (color online). (a) Efficiency of the STIRAP-like PAP
with unshaped laser pulses in dependence on the interpair delay
T and the intrapair delay T. Positive T indicates that the
pump comes after the dump; i.e., the population remains in the
excited wave packet until the next dump comes (suffering more
spontaneous emission losses). (b) Section of the density plot (a)
taken at T 1310:62 ps [blue dotted line in (a)]. (c) Fourier
transform of (b). The green dotted line marks the beat frequency
2
between E1
2 and E2 .
carrier-envelope frequency difference according to f0P f0D !Raman =2 Nfrep . The plot shows a remarkable
landscape of vertical and horizontal features, where some
peaks reach 80% transfer efficiency. Note that no population is transferred to any of the a3 or X1 states,
except j1i and j3i, or to any of the intermediate states. In
the spectral domain, either a vibrational eigenstate guides
one of the interfering AP pathways, or it is simply missed
by the narrow spectral selectivity of the frequency combs.
In the time domain, destructive Ramsey-type interference
prevents transitions to undesired levels.
To elucidate some of the structure observed in Fig. 5,
consider T expressed in frequency as a linear spectral
phase of the dump comb with respect to the pump comb.
Accordingly, the phase of a specific Raman path at detuning i from the carrier frequency is ’i i T d i , where d i is the phase acquired during the fast
pump and dump transitions due to the relative phase of the
dipoles associated with this path. The vertical lines in the
image correspond therefore to a T value where one or
several teeth are in one-photon resonance with Raman
paths. Changing T at this T reveals interference between these paths. A Fourier transform with respect to T
reveals the spacing between the intermediate levels, as
shown in Figs. 5(b) and 5(c), where mostly two Raman
1 and E2 11190 cm1 ,
paths, via E1
2 11145 cm
2
contribute, and the Fourier transform of the transfer effi2
ciency peaks at the beat frequency between E1
2 and E2 .
Horizontal lines, on the other hand, occur when T hits a
special value, where for many participating Raman transitions ’d i i T ’0 ; i.e., T is matched to the
vibrational dynamics. In this case the transfer peaks even if
there is no coherent accumulation in the intermediate
levels, as was observed already in [7]. Since the match to
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the dynamics is only approximate, the lines are not perfectly horizontal. In the best case scenario every dump
pulse transfers into j3i exactly all the population excited
to j2i from j1i by the preceding pump [7].
With some preliminary knowledge of the initial and
target state’s energies, the range of frequencies and the
field strengths needed (but not of the exact values of
intermediate energies and transition dipoles), one can
take trains of unshaped pump and dump pulses, and scan
T and T to find efficient PAP transfers. In an analogy
with 2D Fourier spectroscopy [18], a theoretical analysis of
the 2D scan of the transfer efficiency may be able to
provide the full spectroscopic information. Further, one
can experimentally optimize the pulse shapes in order to
maximize the transfer fidelity.
We thank V. Milner, I. Thanopulos, C. Koch, and
S. Kotochigova for discussions and consultations. The
work at JILA was supported by NIST and NSF.
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[19] From NIST Atomic Spectra Database, (http://physics.nist.
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23 3 a:u:
023601-4